What I would tell someone is this:
Tae Kwon Do is the modern Korean martial art that originated immediately after WWII. Developed by the coming together of several masters and the schools they founded, Tae Kwon Do draws on native Korean fighting techniques, as well various Chinese and Japanese influences that the school founders had studied while overseas before and during WWII.
Tae Kwon Do is a hard striking and blocking style, particularly emphasizing high section standing, jumping, spinning, and flying kicks. However, it does incorporate a number of hand techniques in a vast array of blocking and hitting methods.
Tae Kwon Do incorporates forms, free fighting, self defense, breaking practice, as well as traditional Korean and Oriental philosophy.
Modern Tae Kwon Do can be divided into several organizations, each with its own forms, free fighting rules, and history. These organizations are:
1. The World Tae Kwon Do Federation based in Seoul South Korea and headquartered in the Kukkiwon or Central Gymnasium. One of the leading proponents of full contact Olympic sparring, the WTF practices the Palgue/Taegeuk/Koryo forms.
2. The International Tae Kwon Do Federation founded by Gen. Hong Hi Choi, now deceased. Gen. Choi was a pioneer of early Tae Kwon Do and developed many of the Chang Hon forms used by the ITF. He left Korea due to organizational and political differences and based the ITF originally in Canada. Currently, a schism caused by his death has split the ITF into three separate factions each claiming authenticity.
3. The American Tae Kwon Do Association, founded by Haeng Ung (Hank) Lee, now deceased. The ATA can best be described as an offshoot of the ITF, since that is where GM Lee originated. However, after coming to America to teach, Lee developed his own forms (Songahm forms), which he patented, and based his ATA on his interpretations of Korean Tae Kwon Do and culture.